A dominating family on $\omega^\omega$ is a set $\mathcal D \subset \omega^\omega$ such that for every $f \in \omega^\omega$ there exists $g \in \mathcal D$ such that $f<^* g$ (that is, $f(n)<g(n)$ for all but a finite number of $n$'s). The smallest cardinality of a dominating family is $\mathfrak d$, and it is well known that $\omega_1\leq \mathfrak d\leq \mathfrak c$ and that it is consistent that $\mathfrak d$ may be almost everything between $\omega_1$ and $\mathfrak c$. I know that this is an important combinatorical concept and that it is widely used/studied.

I am interested in reading about some related questions. One may also define dominating families on $\omega^{\omega_1}$ and on $\omega_1^{\omega_1}$, and then define $\mathfrak d_{\omega_1, \omega}$ and $\mathfrak d_{\omega_1, \omega_1}$ in an analogous manner (so $\mathfrak d_{\omega, \omega}=\mathfrak d$).

I would like to know more about dominating families of higher cardinalities, so my main question is "where may I learn more about them?" Can someone please tell me some references?

Some questions (out of my mind) I would like to explore are:

- It is easy to see that $\mathfrak d\leq \mathfrak d_{\omega_1, \omega}$. However, are they the same? What are the possible relations between them?
- Since $\omega_1$ is regular, it is easy to see that $\omega_2\leq \mathfrak d_{\omega_1, \omega_1}\leq 2^{\omega_1}$. Is it consistent that $\mathfrak d_{\omega_1, \omega_1}<2^{\omega_1}$?
- Is it consistent that $\mathfrak d_{\omega_1, \omega}=\omega_1$?